What is our universe expanding into?

First, a quick note to explain why this question is being posted in the "Beyond the Standard Model" forum, as opposed to the relativity forum. Many studies of relativity discuss the expansion of our universe, but I haven't seen many which explicitly target the question of what our universe is expanding into. In contrast, when people like Alex Vilenkin and Alan Guth speculate about these topics, they do so in the context of GUTs, superstring theory, and linkages between gravity and QM.

Question: What is our universe expanding into? Many people describe the Big Bang as an expansion, but simultaneously write that the universe is not expanding into anything. They say that it is mathematically possible to describe space as an expanding manifold, without that manifold expanding into a higher dimensional space, and also (perhaps a simpler explanation) without expanding into a prior 3D space. But they don't offer any physical interpretation for their claim, which is very unphysics-like.

"What is the Universe expanding into?" This question is based on the ever popular misconception that the Universe is some curved object embedded in a higher dimensional space, and that the Universe is expanding into this space. This misconception is probably fostered by the balloon analogy which shows a 2-D spherical model of the Universe expanding in a 3-D space. While it is possible to think of the Universe this way, it is not necessary, and there is nothing whatsoever that we have measured or can measure that will show us anything about the larger space. Everything that we measure is within the Universe, and we see no edge or boundary or center of expansion. Thus the Universe is not expanding into anything that we can see, and this is not a profitable thing to think about. Just as Dali's Corpus Hypercubicus is just a 2-D picture of a 3-D object that represents the surface of a 4-D cube, remember that the balloon analogy is just a 2-D picture of a 3-D situation that is supposed to help you think about a curved 3-D space, but it does not mean that there is really a 4-D space that the Universe is expanding into."​

This answer is common, but enormously unhelpful. Are we really to believe that the volume of our universe is constant? If so, then how could the space between distance galaxies be expanding? The universe cannot be expanding, while simultaneously remaining constant in volume, having rulers with a constant size.

The quote also seems to be misleading: Many physicists openly state that our universe did appear in a pre-existing 3D space of some sort. (Or perhaps a higher dimensional space.) Many cosmologists posit that other Big Bangs likely have occured in this same huge 3D (or more D) space. These other Big Bangs would then create many other universes, and we probably have no way of observing these other universes.

So here is my observation: When physicists talk about multiple universes, they never make the argument that space expands without the need for a larger volume for it to expand into. They openly state that we are expanding into some larger domain. I've seen articles by Alan Guth (known for inflation) and Max Tegmark (known for positing many types of multiverses) which seem to openly state that the Big Bang occured (better: is still occuring) in some 3 or more dimensional space.

Only when physicists discuss our universe alone do they fall back on the "expansion without expansion" argument.

Sometimes I get the idea that many physicists were brainwashed by their teachers into thinking that such questions were simply not allowed. This has happened before: From the 1920s to the 1970s most physicists were brainwashed into believing that Quantum Mechanics requires no interpretation. The Copenhagen interpretation, which required magic observors that mystically collapsed the waveform, was the only "interpretation" allowed, and it basically said nothing about the ontological reality of the universe. Hundreds of physicists were brow-beated into the "shut up and calculate" approach.

As we now know, that approach is now rejected. Today we admit that QM requires some kind of interpretation, and this has become an active field .

So is it still ok to say that that our universe is DEFINITELY NOT expanding into any 3D space, or into any higher dimensional space? If so, then what physical interpretation allows the expansion of our universe, yet without a change in volume?

Or, as I suspect, is the old answer based on outdated dogma? Are physicists now comfortable in saying that our universe is expanding into some space (even though we cannot currently detect it?)

Consider the recent paper by Guth and Vilenkin, "Eternal inflation, bubble collisions, and the persistence of memory"

A ``bubble universe'' nucleating in an eternally inflating false vacuum will experience, in the course of its expansion, collisions with an infinite number of other bubbles. In an idealized model, we calculate the rate of collisions around an observer inside a given reference bubble. We show that the collision rate violates both the homogeneity and the isotropy of the bubble universe. Each bubble has a center which can be related to ``the beginning of inflation'' in the parent false vacuum, and any observer not at the center will see an anisotropic bubble collision rate that peaks in the outward direction. Surprisingly, this memory of the onset of inflation persists no matter how much time elapses before the nucleation of the reference bubble.

..
Question: What is our universe expanding into? Many people describe the Big Bang as an expansion, but simultaneously write that the universe is not expanding into anything. They say that it is mathematically possible to describe space as an expanding manifold, without that manifold expanding into a higher dimensional space, and also (perhaps a simpler explanation) without expanding into a prior 3D space. But they don't offer any physical interpretation for their claim, which is very unphysics-like.

"What is the Universe expanding into?" This question is based on the ever popular misconception that the Universe is some curved object embedded in a higher dimensional space, and that the Universe is expanding into this space. ...​

This answer is common, but enormously unhelpful. Are we really to believe that the volume of our universe is constant? ...

The quote also seems to be misleading: ... Many cosmologists posit that other Big Bangs likely have occured in this same huge 3D (or more D) space. ...Only when physicists discuss our universe alone do they fall back on the "expansion without expansion" argument.

Sometimes I get the idea that many physicists were brainwashed by their teachers into thinking that such questions were simply not allowed. ...
Any thoughts would be greatly appreciated.

No one that I know is trying to convince anybody that the volume of the universe is constant. Why do you think this?

You could have asked this in Cosmology. The question has come up repeatedly there and the regulars are used to responding. Maybe the mods will move the thread to Cosmology Forum.

there are some good threads in Cosmology about what is meant by saying space "expands". It is a shorthand expression that tends to confuse people. there has been a great deal of discussion about this and quite a few of the regulars have worked hard resolving confusions of the kind you bring up.

"space expands" is a shorthand expression for something that happens to the distance-function or METRIC which defines spacetime geometry, it means DISTANCES EXPAND, and in standard cosmology they don't all expand uniformly, only approximately so.

largescale distances are currently growing at a rate of roughly 1 percent every 140 million years, but it isnt UNIFORM. there are identifiable patches where expansion is much slower---the evolution of the metric is governed by Einstein's equation and the distribution of matter.

because distances are increasing, roughly at the rate I said, the corresponding VOLUMES ARE INCREASING currently at roughly 3 percent every 140 million years. The whole shebang might be infinite and therefore not have a well-defined volume. But certainly if the volume is finite then it is increasing. This increase is only approximately uniform at large scale. The rate I'm giving you is more an estimated average.
In any case this increase in volume does not require our space to be surrounded by a larger space!

Nobody brainwashed me, and it never occurred to me that space could only expand if it were surrounded by some larger empty space into which it "grows". That is an extraneous "womb-like" addition to the picture---there doesnt have to be any space "outside" our space

MAYBE YOU ARE CONFUSED BY THE IDEA OF BOUNDARY. Space can be finite volume and yet have no boundary. We have some threads about that in the Astronomy section.

As a footnote on this, some multiverse philosophers picture space has having various regions expanding at different rates----so we are in one region and there are other regions, but instead of solving your puzzle, that makes it worse. Now you have an even larger space expanding in an even more irregular way, with one patch expanding at this rate and another at that rate!

…"space expands" is a shorthand expression for something that happens to the…METRIC which defines spacetime geometry, it means DISTANCES EXPAND...

largescale distances are currently growing…the evolution of the metric is governed by Einstein's equation and the distribution of matter…because distances are increasing…VOLUMES ARE INCREASING

People often erroneously believe that the increasing distances between galaxies means that space is in some sense expanding. One should be careful and say that only under the influence of a positive cosmological constant can space legitimately be said to expand.

No one that I know is trying to convince anybody that the volume of the universe is constant. Why do you think this?

Because a three dimensional volume cannot increase in volume unless it expands into a larger, surrounding volume. This is true by definition. But everyone here, and most physics books, state that the universe is expanding without any expansion into anything. You seem to to say the same thing yourself here, but I don't understand this.

Ironically, I would understand the inverse claim: What if someone said "The volume of the universe is constant, but its surface area is increasing, and may become infinite!" This claim makes sense; Consider the fractal nature of a shoreline. The area of land on an island is constant, but its perimeter approaches infinity as we examine the shore on smaller and smaller scales. Similarly, a basketball - or universe - can stay constant in size, and yet have increasing surface area as the surface becomes more bumpy (especialy if it becomes bumpy in a fractal fashion.)

There are some good threads in Cosmology about what is meant by saying space "expands". It is a shorthand expression that tends to confuse people. there has been a great deal of discussion about this and quite a few of the regulars have worked hard resolving confusions of the kind you bring up. "space expands" is a shorthand expression for something that happens to the distance-function or METRIC which defines spacetime geometry, it means DISTANCES EXPAND, and in standard cosmology they don't all expand uniformly, only approximately so. ...because distances are increasing, roughly at the rate I said, the corresponding VOLUMES ARE INCREASING currently at roughly 3 percent every 140 million years. The whole shebang might be infinite and therefore not have a well-defined volume. But certainly if the volume is finite then it is increasing. This increase is only approximately uniform at large scale. The rate I'm giving you is more an estimated average.

Nobody brainwashed me, and it never occurred to me that space could only expand if it were surrounded by some larger empty space into which it "grows".

What about middle school grade geometry? A balloon cannot - by definition - ever increase in volume, unless it expands into a larger volume. As the volume of a balloon in a room increases, the volume of the surrounding room outside the ball is less. How could this idea not occur to anyone? Perhaps there is something subtle that you're assuming I know, which you are not making explicit?

MAYBE YOU ARE CONFUSED BY THE IDEA OF BOUNDARY. Space can be finite volume and yet have no boundary. We have some threads about that in the Astronomy section.

No, the problem of boundaries doesn't bother me. Either people here don't understand my question, or people here have agreed that balloons increase in size without getting bigger. And this makes no sense.

Let me give you a quote from Leon Lederman. He won the Nobel Prize in Physics, so he is clearly not crazy. This is from "From Quarks to Cosmos", Chapter 6: Page 176 "The New Inflation: Bubble Universes". Lederman starts by mentioning the bubble universe ideas of Andre Linde, Paul Steinhardt and Andreas Albrecht:

...The concept that the universe is a single bubble has tremendous implications: There may be many bubbles out there, all of which could be other universes, completely disconnected from ours. This type of solution involving multiple bubbles, with our universe being a single bubble, was also proposed by Richard Gott of Princeton ...There may be more universes than we ever contemplated, but we will have no way of reaching them....The type of space between bubble universes is not normal physical space at all, but a space where all the forces are unified; a region where protons decay instantaneously, quarks and leptons interchange freely, and normal matter does not exist.​

In a 1994 Scientific American article, "The Self Reproducing Inflationary Universe" Andre Linde states that our universe may be just one bubble in a larger space:

...Such complexities in the scalar field mean that after infation the universe may become divided into exponentially large domains that have different laws of low-energy physics. Note that this division occurs even if the entire universe originally began in the same state, corresponding to one particular minimum of potential energy. Indeed, large quantum fluctuations can cause scalar fields to jump out of their minima. That is, they jiggle some of the balls out of their bowls and into other ones. ...If this model is correct, then physics alone cannot provide a complete explanation for all properties of our allotment of the universe. The same physical theory may yield large parts of the universe that have diverse properties. According to this scenario, we find ourselves inside a four-dimensional domain with our kind of physical laws...​

I have come across numerous other statements from physicists contemplating bubble universes. All of them have one thing in common - they admit that our universe is expanding in a larger 3D space (or higher dimensional, possibly), and they don't see any problems with this at all.

The real problem is twofold, both parts odd and inexplicable to me:

(A) Most physcists have no idea that this point of view that our universe exists in a larger volume, and is expanding into it - even exists! Somehow this common sense view is unknown?

(B) Many physicists insist that our universe is getting larger, yet not expanding into any space, thus simulataneously insisting that our expanding universe is constant in volume. (And note that I am talking about volume, not surface area. This isn't a boundary issue.)

As I said in my initial post, it seems like people think that questions about anything outside our universe is forbidden, and thus no outside must exist. But Leon Lederman, Richard Gott and Andre Linde are not crazy when they write about this. They could be wrong about the existence of other universes, but they are not wrong to merely state that universes occupy some sort of volume, and our universe is expanding into something.

With these quotes in mind, is my initial question more clear? I may be missing something subtle here, but I don't know what.

Perhaps the Big Expansion, rather than just Big Bang, might shift the emphasis to a focus on an expanding manifold. the latter being a set of pts with a local structure; the most primative being topology. what is topology? the study of the continuum. what is continuum? an inbetweeness quallity in the matching of sets. what is the minimum number of elements of a set, in order to have an inbetweenness? 3 elements! Recommend R. Penrose Road to Reality book- nice read emphasizing manifolds. So in consideration of the question: perhaps the universe could be considered as expanding into the antithesis of it's nature - that is, the antithesis of quanta and continuum. but then how might one test this? Assume the opposite, and reductio ad absurdum perhaps. after all such modern day 'ethers' have been ruled too cumbersome and unlikely, perhaps what remains, the simplest case, might seem not so unreasonable - Sherlock Holmes

Because a three dimensional volume cannot increase in volume unless it expands into a larger, surrounding volume. This is true by definition.

I haven't bothered to read the rest of your screed but, if this statement forms an assumption upon which the rest of your post is based, then the rest of your post is incorrect, by definition.

I suspect that your problem is that you are trying to apply "common sense" to this problem, without realising that cosmological expansion is a concept so far removed from common sense intuition as to be understandable only when one uses mathematics to analyse the problem.

Look, to see why what you're saying is patent nonsense, consider the following. If you have some manifold [itex]M[/itex] with positive definite metric [itex]g_{ij}[/itex], then the volume of the manifold is, by definition,

[tex]\textrm{Vol}(M,g) = \int_M d^3x\,\sqrt{g}[/tex]

where [itex]g\equiv\det(g_{ij})[/itex]. Obviously, if you take [itex]M[/itex] to be something like an asymptotically flat manifold then [itex]M\to\infty[/itex]. However, the sorts of manifolds that we are interested in in cosmology are always compact manifolds, without boundary. For example, a common choice in cosmology is to take [itex]M=S^3[/itex], in which case we obtain

[tex]\textrm{Vol}(S^3,g)\propto \pi^2r^3[/tex]

where [itex]r[/itex] is an areal radial coordinate. Now suppose that your claim about the volume of such three-manifolds is true. If it really is true that a three-manifold cannot change its volume without "expanding" or "contracting" into or out of some higher-dimensional space then, if we study three manifolds on their own without any mention of them being embedded in a higher-dimensional space, the notion of volume of such manifolds must be intrinsically meaningless. Put another way, your claim translates to the statement that expressions such as

[tex]\textrm{Vol}(M,g)=\int_M d^3x\, \sqrt{g}[/tex]

are meaningless unless we assume that [itex]M[/itex] is embedded in a higher-dimensional manifold.

This is, however, clearly an absurd thing to say. The volume of a given manifold, if it is finite, is a concept which is completely independent of any embedding of that manifold into a higher-dimensional manifold; it is a concept which is dependent solely upon the metric one uses and, at a deeper level, the topology of the manifold.

You are falling into the very common misconception. If the universe is everything, how can there be something outside it?

No, no one is claiming the universe's volume is constant. No, there does not have to be a 'larger universe' for our universe to expand into. Yes, this is a difficult concept to accept.

Oh boy, this is just not so. The universe, by definition, is not everything. Rather, that used to be the definition, but this definition has changed significantly in the last 20 years.

Modern day physicists and cosmologists define "the universe" as only being everything we know of, created from what we observe as the Big Bang. This includes everything visible within our local Hubble volume. It almost certainly includes many other Hubble volumes, perhaps dozens more, perhaps millions more. And the laws of physics within these surrounding Hubble volumes are, as far as we can discern, identical to the laws of physics within our own.

What, then, do physicists define as outside our universe?

(A) Everything in our universe - even millions of "Hubble volumes" in total volume, could be just one small part of an even larger ginormous mega-universe. Due to inflation, some areas unimagineably far away could have different constants of nature, and different laws of physics. For all intents and purposes these regions would constitute separate universes.

(B) Our universe began from a Big Bang in an empty void of some sort, whose nature is currently undefined (although see the quotes I gave from Leon Lederman to give an idea of what such a void could be like.) However - and this is key - other big bangs could occur that are spatially unconnected to our own. There could be millions of spatially discrete universes with no overlap at all. There could be an infinite number.

(C) Our universe is three dimensional. But our entire 3D universe may be a threedimensional membrane (often called a "brane") in a larger, higher dimensional space (often called "the bulk"). And many other 3D universes may exist within the bulk. These universes would likely be spatially disconnected, not just in three-space, but in other dimensions as well. (In principle, though, some physicists believe that wormholes may connect these universes.)

(D) There are many more ways that other universes can exist. I refer you to the recent writings of Max Tegmark.

I haven't bothered to read the rest of your screed but, if this statement forms an assumption upon which the rest of your post is based, then the rest of your post is incorrect, by definition.

This is too close to an ad homenim remark, and what makes it worse is that it also attacks the views of the eminent physicists that I have named and quoted here. You seem unfamiliar with the concepts they describe, so maybe you should "bother to read the rest".

You astonishingly admit that you won't even read the question, and criticise me on the basis of the first paragraph of a series of queries, with quotes from eminent physicists.

BTW, you can't use math to prove that balloons don't expand. They do. Even big balloons, like our universe. You are confusing a metric, a mere tool, for the physical objects which the math is being used to describe. Look, the laws of physics that you and I write down on paper are not really the universe! All "laws" of physics are merely human conventions, mere convenient tools to describe the actual universe. All laws - even general relativity - are only applicable within a limited domain. Outside this domain our laws fail, or become meaningless by definition.

You described a set of laws to me that were written with the assumption that nothing exists outside our known universe - how then can you conclude from them that nothing exists outside our universe? You are merely restating your premise.

Look at the papers in the ArXiv over the last ten years. Many physicists, including Nobel Prize winners, no longer make this mistake. They have written papers in which they posit that our universe is embedded within a larger volume. Now, they could be incorrect (sure!) but you must not say that they are confused and ignorant.

and what makes it worse is that it also attacks the views of the eminent physicists that I have named and quoted here. You seem unfamiliar with the concepts they describe, so maybe you should "bother to read the rest".

I don't need to read the rest when you make a statement such as:

Robert100 said:

What about middle school grade geometry? A balloon cannot - by definition - ever increase in volume, unless it expands into a larger volume.

It's obvious to anyone who understands the subject that you're trying to discuss that using this sort of analogy to produce "predictions" about the behaviour of cosmological expansion is a non-starter.

Look, let's blow your line of thinking out of the water with a simple example (by simple, I mean that it should be intelligible to anyone who has taken an introductory course in differential geometry). Suppose that you have some smooth, orientable, simply-connected three-dimensional manifold [itex]M[/itex], possibly with non-empty boundary [itex]\partial M[/itex]. Moreover, suppose that [itex]M[/itex] has positive definite metric [itex]g_{ij}[/itex], [itex]i,j=1,2,3[/itex]. As I said earlier, the volume of [itex]M[/itex] is given by

[tex]\textrm{Vol}(M,g) = \int_M d^3x\, \sqrt{g}[/tex]

You can, if you wish, think of this as the volume of your balloon. Notice that I have made no reference whatsoever to [itex]M[/itex] being embedded in a higher-dimensional space. You claim that the volume of this balloon cannot change. I claim it can. To see why I'm right, take a smooth, strictly positive function [itex]\phi[/itex] on [itex]M[/itex] and define a new metric [itex]\overline{g}_{ij}[/itex] according to

[tex]\overline{g}_{ij} = \phi^4g_{ij}[/tex].

Then, if we calculate the volume of [itex]M[/itex] with this new metric we obtain

Then comparing [itex]\textrm{Vol}(M,g)[/itex] and [itex]\textrm{Vol}(M,\overline{g})[/itex] we have the blindingly obvious result that in general,

[tex]\textrm{Vol}(M,\overline{g})\ne \textrm{Vol}(M,g)[/tex]

In other words, the volume of [itex]M[/itex] has changed now that we have used a different metric to measure it with. Nowhere have we made reference to a higher-dimensional space: we have dealt solely with the three-dimensional "balloon". The analogy to cosmological expansion is then simple. Since general relativity predicts that cosmological evolution of a spacetime (which is necessarily compact without boundary) will change the metric, the volume of the associated manifold will, in general, change over time.

just to put Russ Watters' phrase "difficult to accept" in historical perspective I'd say that what shoehorn has just written goes back to a realization by Bernhard Riemann in 1850.

this was that you could study the geometry of a space from inside, using the distance function---the metric---corresponding to the ordinary process of measuring distances

Russ Watters, shoehorn please correct me if I am wrong about the history

Bernhard Riemann was kind of a Mozart of mathematics, his creative insight back in 1800s set the stage for much exciting 20th century discovery.

He realized that you could define the volume of a space whether or not that space was embedded in some larger space. Shoehorn wrote down the integral. the space doesnt even have to have symmetry or rectilinear coordinates.

The phrase "difficult to accept" is both true and not true. It is obviously true in the sense that the Greeks were already studying 3D geometry in 200 BC and they were very clever but their solid 3D bodies were all embedded in a larger space and you could look at them "from the outside"----so the idea of being able to define and study volumes and all sorts of geometry things "from the inside only" was sufficiently unintuitve to the human (monkey-evolved) brain that IT TOOK OVER 2000 YEARS before Riemann got the idea to do it.

But on the other hand once it is in our culture (since 1850) it is a very easy and obvious idea. Of course you can measure volume from the inside!
You can even measure curvature from the inside just by checking the sum of internal angles of a triangle. You can do all sorts of geometry without reference to any external viewpoint.

...
Look at the papers in the ArXiv over the last ten years. Many physicists, including Nobel Prize winners, no longer make this mistake. They have written papers in which they posit that our universe is embedded within a larger volume. Now, they could be incorrect (sure!) but you must not say that they are confused and ignorant.

Robert I urge you to stop the misleading practice of dragging "Many physicists, including Nobel Prize winners" into this.

Of those you might name, I dare say none would agree with you.

Many people study situations where the space we know is included in some larger space and they do that for many interesting reasons. Nobody that I ever heard of does that because he thinks that in order for space to expand it has to be contained in a surrounding space.

That idea, your idea that an expanding volume must be surrounded, is a crackpot idea. You would not get any reputable physicist to listen to it.
You are imposing on people here at PF by persistently implying that your idea is supported by Nobel prize winners etc.

Of course plenty of people speculate about a surround of some sort but I challenge you to provide a link to a passage of text where even one reputable physicist asserts that expanding volume is inherently impossible without a surround.

Rubbish. I don't need to read the rest when you make a statement such as...

Anyone who tries to answer a question without fully reading the question is foolhardy . Anyone who is proud of this is rude. I've never taken that approach with you, or with anyone on this forum. Most importantly, I have never once taken this attitude with any of my own students. Not surprisingly, I don't appreciate (or even understand) this angry treatment. From whence does your anger towards me arise?

It's obvious to anyone who understands the subject that you're trying to discuss that using this sort of analogy to produce "predictions" about the behaviour of cosmological expansion is a non-starter...

You may be very well versed in math, but you don't seem to know much about the meaning of physics. It certainly isn't about math. It is about ideas. In any case, if we can't describe something without tensor calculus, then we likely don't understand it! That is not an ad homenim remark. Richard Feynman went further: He is known for saying that if you can't describe ideas in physics to high school kids, then you really don't understand these ideas yourself. (I think he actually said sixth graders.)

This key: You can actually explain quantum mechanics, special relativity, and even much of general relativity without math at all! And not just silly hand waving using terms. You can really describe these subjects well. And with high school geometry, albegra, trig, and a bit of high school calculus, you can explain tremendous amounts of these subjects, and even discuss subtle ideas such as the Bell inequality.

Your approach is to be rough, throw graduate physics integrals around, and demand to be taken at your word. As a teacher by profession, I can tell you that this is not the most effective pedagogy. Consider this statement of yours:

Look, let's blow your line of thinking out of the water with a simple example (by simple, I mean that it should be intelligible to anyone who has taken an introductory course in differential geometry).

Well, I have not taken differential geometry. Neither have most people who are interested in learning about relativity, cosmology and the like. Your argument boils down to "Become a full time physicist or just believe my assertations without any reasoning!" Now, I'm sure this is not what you mean, but this is the mesage you give. Maybe I am being too sensitive, but as a high school physics teacher this is an important issue for me.

Albert Einstein found a way to explain general and special relativity for laypeople. Richard Feynman actually explained quantum electrodynamics to lay people (and many practicing physicists admit that they continue to learn from his "popular" texts.) So is it so wrong for me to believe that someone might point me to some sort of explanation that doesn't require me to quit my job and start gradudate school in physics?

I find it hard to believe that no possible explanation or analogy can be constructed.

On to the math:

Then, if we calculate the volume of [itex]M[/itex] with this new metric we obtain

Then comparing [itex]\textrm{Vol}(M,g)[/itex] and [itex]\textrm{Vol}(M,\overline{g})[/itex] we have the blindingly obvious result that in general,

[tex]\textrm{Vol}(M,\overline{g})\ne \textrm{Vol}(M,g)[/tex]

In other words, the volume of [itex]M[/itex] has changed now that we have used a different metric to measure it with. Nowhere have we made reference to a higher-dimensional space: we have dealt solely with the three-dimensional "balloon".

Is this saying that the volume changes merely by looking at the same volume from a different reference frame? I don't understand the physical meaning of this change.

The analogy to cosmological expansion is then simple. Since general relativity predicts that cosmological evolution of a spacetime (which is necessarily compact without boundary) will change the metric, the volume of the associated manifold will, in general, change over time.

Robert I urge you to stop the misleading practice of dragging "Many physicists, including Nobel Prize winners" into this. Of those you might name, I dare say none would agree with you. Many people study situations where the space we know is included in some larger space and they do that for many interesting reasons. Nobody that I ever heard of does that because he thinks that in order for space to expand it has to be contained in a surrounding space.

Look, I may be incorrect in interpreting what I read. Feel free to say so and explain why. But Marcus, please do not imply that I am trying to mislead people. I don't understand where this charge is coming from.

That idea, your idea that an expanding volume must be surrounded, is a crackpot idea. You would not get any reputable physicist to listen to it.

Marcus, I literally have no idea what you are talking about! In fact, isn't the opposite true? Didn't you previously write that not only is it not a crackpot idea, but in fact this is a common sense truth that everyone accepted for the last 6,000 years of human civlization, and only recently began to be questioned? (And as one person here seems to imply, this new idea can only be understood by people mastering grad school physics?) So why are you so surprised and angered that someone would have this sort of question?

When people ask about addition of velocity in the relativity forum here, and ask about traditional, Galilean relativity, do you usually attack them as "crackpots"? That's not how we teach physics. In fact, that is not how we teach anything.

You are imposing on people here at PF by persistently implying that your idea is supported by Nobel prize winners etc.

You are conflating two separate ideas. I did not make this precise claim.

In any case, after a year of participating here, I am very surprised, and at a loss to explain these seemingly angry reactions.

Perhaps you can more charitably understand my concerns if I phrase it this way:

(a) is the math proving that no outside can possibly exist?

(b) is the math, at least, strongly suggesting that no outside can possibly exist?

(c) is this math actually silent on the issue? That is, the math only describes the changing volume of the universe, without need for us to discuss any larger volume it could conceivably be embedded in?

As far as I can tell, people here are implying (c), yet this doesn't seem to contradict anything I have speculated about. Or if you think that it does contradict what I have asked about, could someone point me to an explanation (book, website, whatever) that doesn't require tensor calculus?

You may be very well versed in math, but you don't seem to know much about the meaning and message of physics. It certainly isn't about math.

To a large part it is about the maths. If you don't understand the maths that describe a theory it is impossible to comprehend the meaning (at least in full). What shoehorn has done in this thread is not just state maths without meaning, but he's tried very patiently to explain the meaning of the maths. I implore you to have another read of his posts, the meaning you seek is there, but you can't divorce physics from the maths no more than you can divorce Shakespeare from the English language.

Well, I have not taken differential geometry. Neither have most people who are interested in learning about relativity, cosmology and the like. Your argument boils down to "Become a full time physicist or just believe my assertations without any reasoning!" Now, I'm sure this is not what you mean, but this is the mesage you give. Maybe I am being too sensitive, but as a high school physics teacher this is a key point for me. (We don't need grad school GR maths to teach at this level!)

With respect I think most people who are into relativity, cosmology and the like do end up learning some differential geometry since otherwise you simply cannot read anything other than pop sci. The worst thing you can ever try and do is to do 'science by analogy' for instance philosophising about quantum mechanics by thinking about cats in boxes or GR by thinking of rubber sheets. These things are analogies to help you understand the current ideas but cannot be used to try and pick holes in a theory or extend the ideas simply because they are not the theory, they are merely analogies.

Sure Feynman and others have stressed that you should be able to explain anything in physics to the general public, but that most certainly does not mean that having had something explained at that level to you means you can turn the tables and extend the physics. It does not work both ways.

I'd attempt to address the specifics of the issue your having, but unless we can agree on the acceptable process by which we can do science we're just punching each others smoke as you are shoehorn have done thus far.

L
Perhaps you can more charitably understand my concerns if I phrase it this way:

(a) is the math proving that no outside can possibly exist?

(b) is the math, at least, strongly suggesting that no outside can possibly exist?

(c) is this math actually silent on the issue? That is, the math only describes the changing volume of the universe, without need for us to discuss any larger volume it could conceivably be embedded in?

As far as I can tell, people here are implying (c), yet this doesn't seem to contradict anything I have speculated about.

(c) the math is silent. Occam is not silent. He says entities should not be needlessly multiplied. But you can always choose to ignore Occam

What infuriated me was that you seemed to be claiming that a bunch of physicists including a Laureate were on your side.

You have been pretty consistently maintaining what I think is an illogical proposition, namely
"Robert's theorem" EXPANSION IMPLIES A SURROUND
Robert's theorem can not be proven and I cannot imagine any competent physicist, Laureate or otherwise, who would think it was correct.

I've had it with Robert's theorem.

However I am not bothered by your SPECULATING that there might be a surround. Lots of cosmologists speculate about that. I would guess they are very far from a majority but that is of no consequence. Everybody is allowed to speculate and we aren't deciding things by vote.

Different people imagine a surround for different reasons, some quite far-fetched----elaborate inflations scenarios, whole universes on a brane floating in higher dimensional space, braneworlds colliding with each other.
To some extent people latch on to these ideas because THEY APPEAL TO SOME PEOPLE'S IMAGINATION, but they also concoct these pictures because they CANT THINK OF A SIMPLER ALTERNATIVE EXPLANATION for basic stuff like the temperature map of the CMB.

Sure Feynman and others have stressed that you should be able to explain anything in physics to the general public, but that most certainly does not mean that having had something explained at that level to you means you can turn the tables and extend the physics. It does not work both ways.